My vet told me to use a livestock dust insecticide that is good for lactating animals. She told me to just apply on the back against hair growth (from tail across spine to neck). Within 2 days my goats stopped scratching. I didn't however apply it to my week old buckling, but he is so actively jumping all over Momma that I am sure some has rubbed off on to him. Hope this helps.
Ivermec either oral, topical or injectable will take care of lice - you dose it weekly three or four times and is safe on kids 4 weeks and older and all adults. Cylence is another good product to rid your herd - but I'm not sure how young you can administer it.
Don't forget to treat the bedding areas with Sevin.
My vet recommended Ovitrol spray for lice. It is a pyrethrin based egg killing spray. I groomed it into the goats' coats when I got them (sprayed on and stiff brushed in) and then have used it once per month this winter just to be sure.
That Ivermection Drench stuff works wonders. I got 5 goats for free from a lady who couldn't care for them any longer, and when they came home, I saw the lice. So I gave the Ivermection to the mothers, because the twins were only 5 hours old, and the other baby was not even a month. I put tons of powder in the stall and on their bedding (and also sprinkled it around there yard. I have a sheep, and she definitely could not get it because she has beautiful wool, and I would be devastated if she got it. After doing all that, they were gone forever.
My goats are a large breed. My vet gave me dectomax injection last summer because we had issues with so many ticks - more than the guineas could keep under control. One injection controls internal and external parasites. Funny thing is only my boer cross buckling had issues with ticks. We never found ticks on any of the other goats but every day I would have to pull off ticks from around his horns. After day in and day out I asked her what I could do for him because nothing was working to keep them off his head and she told me to give him the injection and it would treat lice too. I did. No more ticks for him since that time. None of the other goats have ever needed anything like that. They stay clean and worm free without any aid.
I cannot even imagine the # of quarts/gallons it would take to cover a goat in mayonnaise enough to smother out lice especially deep down in a winter coat of fur. I can image the mess and the yuck and muck that would be created while said goat roams the fields and lays around in a barn stable. Not to mention the smell of warm mayonnaise on a goat.